Question

$\text{If Z is a complex variable, the value of}{\int }_3^{2i}\frac{dZ}{Z}\text{is}$

• A. $0.405+1.57i$
• B. $-0.405-1.57i$
• C. $0.405-1.57i$
• D. $-0.405+1.57i$

Answer 1

The correct option is D $-0.405+1.57i$

$\text{If Z is a complex number, the value of}{\int }_3^{2i}\frac{dZ}{Z}\text{is log}Z{|}_3^{2i}$
$logZ=\frac{1}{2}ln(x^2+y^2)+i{\text{tan}}^{-1}\left(\frac{y}{x}\right)$

$=\text{log}2i-\text{log}3$

$=\text{log}(0+2i)-\text{log}(3+0i)$

$=\left[\frac{1}{2}ln\right(0+4)+i{\text{tan}}^{-1}\left(\frac{2}{0}\right)]-\left[\frac{1}{2}ln\right(9+0)+i{\text{tan}}^{-1}\left(\frac{0}{3}\right)]$

$=\frac{1}{2}ln4+i{\text{tan}}^{-1}\infty -\frac{1}{2}ln9-0$

$=ln(4{)}^{1/2}+i{\text{tan}}^{-1}\infty -ln(9{)}^{1/2}$

$=ln2+i\frac{\pi }{2}-ln3=\left(\frac{2}{3}\right)+i\frac{\pi }{2}$

$=-0.4054+1.57i$